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Estimate State-Space Models with Canonical Parameterization

What Is Canonical Parameterization?

Canonical parameterization represents a
state-space system in a reduced parameter form where many elements
of A, B and C matrices
are fixed to zeros and ones. The free parameters appear in only a
few of the rows and columns in state-space matrices A, B, C, D,
and K. The free parameters are identifiable —
they can be estimated to unique values. The remaining matrix elements
are fixed to zeros and ones.

The software supports the following canonical forms:

Companion form: The
characteristic polynomial appears in the rightmost column of the A matrix.

Modal decomposition form:
The state matrix A is block diagonal, with each
block corresponding to a cluster of nearby modes.

Note:
The modal form has a certain symmetry in its block diagonal
elements. If you update the parameters of a model of this form (as
a structured estimation using ssest),
the symmetry is not preserved, even though the updated model is still
block-diagonal.

Observability canonical form:
The free parameters appear only in select rows of the A matrix
and in the B and K matrices.

For more information about the distribution of free parameters
in the observability canonical form, see the Appendix 4A, pp 132-134,
on identifiability of black-box multivariable model structures in System
Identification: Theory for the User, Second Edition, by
Lennart Ljung, Prentice Hall PTR, 1999 (equation 4A.16).

Estimating Canonical State-Space Models

You can estimate state-space models with chosen parameterization
at the command line.

For example, to specify an observability canonical form, use
the 'Form' name-value pair input argument, as follows:

m = ssest(data,n,'Form','canonical')

Similarly, set 'Form' as 'modal' or 'companion' to
specify modal decomposition and companion canonical forms, respectively.

If you have time-domain data, the preceding command estimates
a continuous-time model. If you want a discrete-time model, specify
the data sample time using the 'Ts' name-value
pair input argument:

md = ssest(data, n,'Form','canonical','Ts',data.Ts)

If you have continuous-time frequency-domain data, you can only
estimate a continuous-time model.

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